Slices, slabs, and sections of the unit hypercube
Jean-Luc Marichal, Michael J. Mossinghoff
TL;DR
This paper derives formulas for the volume of convex bodies formed by intersecting a unit hypercube with various hyperplanes using combinatorial methods, exploring historical context and applications.
Contribution
It introduces new combinatorial formulas for hypercube intersections with hyperplanes, expanding understanding of these geometric volumes.
Findings
Formulas for hypercube-halfspace intersections
Formulas for hypercube-hyperplane intersections
Discussion of historical development and applications
Abstract
Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a halfspace, or with a hyperplane of codimension 1, or with a flat defined by two parallel hyperplanes. We also describe some of the history of these problems, dating to Polya's Ph.D. thesis, and we discuss several applications of these formulas.
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Taxonomy
TopicsPoint processes and geometric inequalities · Mathematics and Applications · Computational Geometry and Mesh Generation